Universal (Meta-)Logical Reasoning: Recent Successes

Abstract

Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.

Full text

Universal (Meta-)Logical Reasoning: Recent Successes
Christoph Benzm¨ullera,b
aFreie Universit¨at Berlin, FB Mathematik und Informatik, D-14195 Berlin, Germany
bUniversit´e du Luxembourg, FSTC, L-4365 Esch-sur-Alzette, Luxembourg
Abstract
Classical higher-order logic, when utilized as a meta-logic in which various other
(classical and non-classical) logics can be shallowly embedded, is suitable as a
foundation for the development of a universal logical reasoning engine. Such
an engine may be employed, as already envisioned by Leibniz, to support the
rigorous formalisation and deep logical analysis of rational arguments on the
computer. A respective universal logical reasoning framework is described in
this article and a range of successful first applications in philosophy, artificial
intelligence and mathematics are surveyed.
Keywords: computational metaphysics, classical higher-order logic,
non-classical logics, automated reasoning
1. Introduction
The quest for a universal reasoning framework is very prominently repre-
sented in the works of Leibniz. He envisioned a scientia generalis founded on
acharacteristica universalis, that is, a universal formal language in which all
knowledge about the world and the sciences can be encoded. A quick study
of the survey literature on logical formalisms suggests that quite the opposite
to Leibniz’s dream has become reality. Instead of a characteristica universalis,
we are today facing a very rich and heterogenous zoo of different logical sys-
tems, and instead of converging towards a single superior logic, this logic zoo is
further expanding, eventually even at accelerated pace. As a consequence, the
unified vision of Leibniz seems farther away than ever before. However, there
are also some promising initiatives to counteract these diverging developments,
and related works on unifying approaches to logic include categorial logic [1, 2],
algebraic logic [3] and coalgebraic logic [4, 5]. While some practical work has
been reported utilizing the algebraic logic approach [6, 7], these approaches
typically have had a strong emphasis on theory only.
IThis research was funded by the German National Research Foundation (DFG) under
Heisenberg grant BE 2501/9 (Studies in Computational Metaphysics) and by Volkswagen
Stiftung under grant CRAP (Consistent Rational Argumentation in Politics).
URL: http://christoph-benzmueller.de (Christoph Benzm¨uller)
Preprint submitted to Science of Computer Programming November 8, 2018

The solution presented here draws on another alternative at universal logical
reasoning: the shallow semantical embeddings (SSE) approach. This approach
has a very pragmatic motivation, foremost reuse of tools, simplicity and ele-
gance. It utilises classical higher-order logic (HOL) [8, 9] as a unifying meta-
logic in which the syntax and semantics of varying other logics can be explicitly
modeled and flexibly combined. Off-the-shelf interactive theorem provers (ITPs)
and automated theorem provers (ATPs) for HOL [10] can then be employed to
reason about and within the shallowly embedded logics.
This survey article summarises and reflects upon the main results of the
research project Studies in Computational Metaphysics (CompMeta), conducted
from 2012 to 2017 at Freie Universit¨at Berlin and Stanford University. In this
project the SSE approach has been further developed and empirical studies
have been conducted in various disciplines, including philosophy, mathematics
and artificial intelligence. In philosophy/metaphysics, for example, an initial
focus has been on computer-supported assessments of modern variants of the
ontological argument for the existence of God, where the SSE approach has
been utilised in particular for automating variants of higher-order (multi-)modal
logics [11]. It is these sort of challenge applications of expressive logical reasoning
that the SSE approach is primarily addressing.
A most relevant aspect from the perspective of computer programming is
that the SSE approach in an elegant and theoretically well-founded manner
strives for maximal reuse of already existing theorem proving technology with
minimal coding effort. The prototype systems that have been implemented in
the course of CompMeta performed surprisingly well in all the conducted case
studies, which provides good evidence for the practical relevance of the approach.
Note that the performance of these implemented systems will (to some extent)
naturally advance in the future without much effort simply because the state-
of-the-art ATPs and satisfiability modulo theories (SMT) solvers it integrates
will further improve in regular cycles.
The article is structured as follows. Section 2 outlines the SSE approach
and discusses its application with a challenge puzzle in epistemic reasoning: the
wise men puzzle. The presented solution puts a particular emphasis on the
adequate modeling of common knowledge. Section 3 presents the motivation
and objectives of the CompMeta project, in which the SSE approach has been
further explored and empirically assessed. The main results of CompMeta are
subsequently summarised and discussed in Sect. 4. Section 5 concludes the
article.
2. The Shallow Semantical Embeddings Approach
HOL has its roots in the logic of Frege’s Begriffsschrift [12]. However, the
version of HOL as addressed here is a (simply) typed logic of functions, which
has been proposed by Church [8]. It provides lambda-notation, as an elegant
and useful means to denote unnamed functions, predicates and sets (by their
characteristic functions). Types in HOL eliminate paradoxes and inconsisten-
cies. Russel’s paradox (the set of sets which do not contain themselves), for

example, which can be formalized in Frege’s logic, cannot be represented in
HOL due to type constraints. For more details and further references on HOL
and its automation we refer to the literature [9, 10]. Very relevant for the work
presented here is that the theory of HOL is well understood [13, 14] and that off-
the-shelf ATPs and ITPs for HOL exists, which can easily be reused. Respective
reasoning systems that are particularly relevant for the SSE approach include
the proof assistants Isabelle/HOL [15] and Coq [16], the ATPs LEO-II [17],
Leo-III [18] and Satallax [19], and the model finder Nitpick [20]. In the running
example discussed below the Isabelle/HOL proof assistant is used. There are
two main reasons for this choice: (i) The powerful graphical interface of Isa-
belle/HOL enables a particularly intuitive interaction with the SSE approach
in which the logic embeddings can be very elegantly displayed and edited, and
(ii) this system, via its Sledgehammer tool [21], integrates powerful first-order
(FO) ATPs and SMT solvers, including E [22], CVC4 [23], Z3 [24], SPASS [25]
and Vampire [26], and it also connects with the HOL ATPs Leo-II and Satallax.
This combination makes it a most suitable environment for conducting mixed
interactive and automated experiments with the SSE approach.
In the remainder of this section the SSE approach will be outlined with the
help of a prominent puzzle in epistemic reasoning: the wise men puzzle (cf. its
discussion in the literature [27, 28]). A particular emphasis and novelty in the
formalisation below is the adequate modeling of the common knowledge of a
set of agents, which is defined as the transitive closure of the agents mutual
knowledge. While the adequate encoding of the notion of transitive closure
poses a challenge for inexpressive knowledge representation frameworks, we here
utilise and showcase a particularly short and elegant solution in HOL (a single
line of code).
2.1. Outline of the SSE Approach
Let L be an object logic of interest, for example, higher-order (HO) modal
logic, which amongst others has prominent applications in metaphysics. Since
our running example below requires the combination of modalities, we will in
fact work with a HO multi-modal logic (HOMML).
The overall idea of the SSE approach is to provide a lean and elegant equa-
tional theory which interprets the syntactical constituents of logic L (in our case
HOMML) as lambda-terms of the meta-logic HOL.
An encoding of HOMML in HOL is presented in Fig. 1. Unlike the traditional
translation approach [29], the connection between HOMML and HOL, i.e. the
equational theory defining the translation is itself formalised in HOL. Moreover,
in contrast to a deep logical embedding, where the syntax and the semantics
of logic L would be formalised in full detail (using structural induction and
recursion), only the crucial differences in the semantics of both, L and HOL, are
directly addressed in the equational theory, while the commonalities are shared
between both logics. HOMML and HOL, for example, share the domain of
individuals. A crucial difference, however, lies in the possible world semantics
on the side of HOMML. Hence, the equational theory defining HOMML in HOL
provides an explicit modeling of this particular aspect of modal semantics. The

Figure 1: Shallow semantical embedding of HOMML in HOL.
central idea of this theory is to associate Boolean valued formulas ϕoof HOMML
with world-predicates (truth-sets characterised as lambda-abstractions) ϕi→oin
HOL, where istands for a reserved type for worlds (lines 5-7 in Fig. 1). The
predicate type i→ois abbreviated as σand αstands for the type i→i→oof
accessibility relations.
To establish our mapping it essentially suffices to equate the classical logical
connectives of HOMML with corresponding world-lifted predicates and relations
in HOL. For example, in line 13 in Fig. 1 the HOMML connective ∧is identified
with the lambda-term λϕ.λψ.λw.ϕw∧ψw, such that ϕ∧ψis mapped to the truth
set {x|ϕw ∧ψw}, which is denoted in HOL by the lambda-term λw.ϕw ∧ψw.
The indexed modal operators 2rof HOMML are identified with lambda-terms
λϕ.λw.∀v.rwv →ϕv, where argument symbol rdenotes an accessibility relation
between possible worlds. We may alternatively say that a parameterised, generic
2-operator is introduced here as λr.λϕ.λw.∀v.rwv →ϕv, such that 2rϕ is
mapped to the truth set λw.∀v.rwv →ϕv (line 21 in Fig. 1). This generic
2-operator can then be instantiated for concrete accessibility relations ras
required.
The mapping of constant symbols and variables from HOMML to HOL is

trivial, since only a type-lifting is required. Most importantly, the mapping
of HOMML to HOL can be defined by a set of non-recursive equations (in
fact, abbreviations), in which the dependency of HOMML formulas on possible
worlds is made explicit, while other aspects and parameters of its semantic
interpretation, such as the underlying semantic domains, remain shared between
HOMML and HOL.
Another interesting and important aspect is that the SSE approach scales
well for FO and HO quantifiers. Analogous to the encoding of the proposi-
tional HOMML connectives, they can be introduced as simple abbreviations of
lambda-terms in HOL as well. The introduction of new binder mechanisms is
not required, since the already existing lambda-binder(s) in meta-logic HOL can
be elegantly reused. This is shown in line 17 in Fig. 1, where the HOL univer-
sal quantifier Π(µ→o)→ois type-lifted to become a HOMML universal quantifier
Π(µ→(i→o))→(i→o). In line 18 convenient binder notation is then defined for
the lifted HOMML quantifier, so that we may write ∀x.φx instead of the less
intuitive variant Π(λx.φx). Type polymorphism is employed in lines 17-18 to
avoid the otherwise required enumeration of such quantifier-defining equations
for different argument types. The existential quantifier for HOMML is intro-
duced analogously in lines 19-20.
The final step is to provide a notion of validity for the type-lifted HOMML
formulas in HOL (see line 25 in Fig. 1): A type-lifted formula ϕ, is valid, denoted
here as bϕc, if and only if the application of ϕto wholds for all worlds w. In
addition, a notion of local validity, denoted here as bϕccw, can be introduced:
bϕccw is true if and only if ϕ cw holds, where cw is an uninterpreted constant
symbol representing the current world.
The presented equations thus characterise a fragment of HOL which, modulo
the above sketched type-lifting, corresponds to HOMML. The faithfulness of this
correspondence, that is, its soundness and completeness with respect to Henkin
semantics, can be established with pen and paper methods [11].1
2.2. Operations on (Accessibility) Relations, including Transitive Closure
Figure 2 presents some useful operations on (accessibility) relations. They
can e.g. be used to elegantly postulate an accessibility relation rto be reflexive,
transitive and euclidean, which are the semantic properties typically associated
with modal knowledge operators 2r(see also line 6 in Fig. 3). The expressivity
of HOL is particularly exploited in the single-line definition of the transitive clo-
sure operation tc in line 14 in Fig. 2, which expresses that two objects (worlds)
xand yare related in the transitive closure (tc R) of a relation Rif and only
if they are related in all transitive super-relations Qof R. Utilising this defini-
tion, ATPs integrated with Isabelle/HOL via Sledgehammer can be employed
to prove some useful lemmata, including the transitivity of the transitive closure
1Further work will investigate whether such faithfulness proofs can eventually be formalised
as well in the approach presented here or whether e.g. a deep embedding is required in this
case.

Figure 2: Operations on (Accessibility) Relations, including Transitive Closure.

of any relation R(line 21) and the symmetry of the transitive closure of any
symmetric relation R(lines 27-28). In the lower window of the GUI in Fig. 2
we e.g. see that the FO ATPs E and SPASS quickly prove the latter lemma,
while the SMT solver CVC4 timesout (the SMT solver Z3 is still running here).
Sledgehammer determines the exact dependencies for the proven conjecture and
it identifies a trusted proof tactic in Isabelle/HOL (here auto), which is capable
of reproving the lemma when the determined dependencies are preselected. The
reconstructed proof utilising the tactic auto is recognised by Isabelle’s inference
kernel. The lemma is then accepted as such by the system.
2.3. Example Application: Wise Men Puzzle
The wise men puzzle, a famous logic riddle whose formalisation has been
studied in some detail e.g. by Baldoni [27], is as follows: Once upon a time, a
king wanted to find the wisest out of his three wisest men. He arranged them
in a circle and told them that he would put a white or a black spot on their
foreheads and that one of the three spots would certainly be white. The three
wise men could see and hear each other but, of course, they could not see their
faces reflected anywhere. The king, then, asked to each of them to find out the
color of his own spot. After a while, the wisest correctly answered that his spot
was white.
An encoding of this epistemic puzzle scenario utilising the SSE approach is
presented in Fig. 3. As an improvement over related work and also over own pre-
vious experiments [30], an adequate modeling of mutual knowledge and common
knowledge is provided in Fig. 3 by following the suggestions of Sergot [31]. The
key idea is to model the knowledge of each wise men, say a, with the help of an
indexed KT45 (=S5) modal operator 2a. We thus introduce three accessibilty
relations a,band c(see line 3 in Fig.3) and instantiate the generic 2-operator
from Fig.1 accordingly to obtain the indexed knowledge operators 2a,2band
2c, one for each wise men (cf. their uses in lines 26, 28 and 30). The accessibilty
relations a,band care constrained in line 6 to obey reflexivity, transitivity and
euclideaness. This ensures that 2a,2band 2care KT45 knowledge operators
as intended. Following Sergot, the mutual knowledge of the wise men a,band
cis introduced next be defining a relation Eabc as the union of the accessibility
relations a,band c(in line 8). However, the corresponding 2Eabc-operator
does not yet qualify as an operator for common knowledge, since it may fail to
be transitive. Hence, another relation Cabc is introduced as the transitive clo-
sure of Eabc. The ATPs integrated with Isabelle/HOL confirm (in lines 12-14)
that Cabc is reflexive, transitive and euclidean, which means that 2Cabc is a
suitable encoding of the common knowledge of the wise men a,band c.
The formalisation of the epistemic puzzle scenario is continued in lines 16-
30 in Fig. 3.2In addition to the already declared constant symbols a,band
2This encoding still abstracts from the temporal dynamics of the scenario, and the adequate
inclusion of such aspects, for example by adopting and integrating an semantic embedding of
dynamic epistemic logic [32], is still ongoing work. The formalisation presented here never-

Figure 3: Wise Men Puzzle.

c, which denote the epistemic accessibility relations of the wise men in the
scenario, two further uninterpreted constants symbols are introduced in line 16
of Fig. 3. Predicate wise is used to identify and denote the set of wise men in
the scenario (see line 18), and the predicate ws expresses whether a wise man
has a white spot. Line 20 states that it is common knowledge of the wise men
a,band cthat at least one of them has a wite spot. Line 22 (respectively, line
24) then postulates that it is common knowledge that if one wise man has a
white spot (respectively, not a white spot), then the other wise men see and
thus know this. This information, which is implicit background knowledge that
is not explicitly stated in the puzzle itself, is nevertheless relevant for solving it.
Note in particular, how the exploitation of the meta-logic HOL here avoids the
otherwise required duplication of these axioms for different combinations of x
and yin the object logic HOMML. Lines 26 and 28 encode the information that
the first two wise men that are asked by the king (they are called aand bhere) do
not know whether they have a white spot. This again is postulated as common
knowledge of the wise men. Then, in line 30, the theorem is formulated that
the third wise men cnow knows that he has a white spot. This theorem can be
proven by the ATPs integrated with Isabelle/HOL via the Sledgehammer tool.
The lower part of the GUI window in Fig. 3 shows that the FO ATPs E and
SPASS succeed, and so does the SMT solver CVC4 (while Z3 is still running).
Moreover, in line 31, a consistency check for the entire formalisation of the
puzzle scenario is performed: the model finder Nitpick computes and presents a
model (not displayed here) that satisfies the presented axioms and definitions.
Note the elegance and minimal effort with which an integrated interactive
and automated theorem proving environment for HOMML has been imple-
mented in Figures 1-3 on top of an existing theorem proving infrastructure
for HOL. In fact, the entire implementation of HOMML and its application to
the wise men puzzle did not require more than 94 line of Isabelle/HOL code,
including commentary. And still, a good degree of automation is achieved,
which significantly benefits from the existing ATPs and SMT solvers already
integrated with Isabelle/HOL via the Sledgehammer tool.
After this brief illustration of the SSE approach we now turn attention to
the CompMeta project and briefly discuss its original motivation, its objectives,
and some relevant preceding work.
3. Ob jectives of the CompMeta Project and Preliminary Work
The CompMeta project has its intellectual roots in the author’s work, con-
ducted with colleagues since the mid-nineties, on the theory and practice of HO
theorem proving (cf. [14, 33, 34] and the references therein), on HO proof assis-
tants (cf. [35, 36] and the references therein), and on their applications in mathe-
theless already elegantly demonstrates some core advantages of the SSE approach, including
the already mentioned appropriate modeling of common knowledge based on the transitive
closure of mutual knoweldge.

matics, artificial intelligence and education (e.g. [37, 38, 39]). These research ac-
tivities inspired first experiments towards the development of a universal (meta-)
logical reasoning framework based on the SSE approach [40, 41, 30, 11]. The
core motivation for the CompMeta project has been to further consolidate these
initial ideas and to assess the approach in empirical studies. The main ob jectives
thus included
1. to further explore the theoretical foundation of the SSE approach,
2. to exemplarily implement the approach for a range of challenge logics in
existing ATPs and ITPs for HOL,
3. to provide evidence for its universal logical reasoning capabilities within
exemplary case studies in metaphysics and beyond,
4. and to educate a new generation of students and researchers to master the
SSE approach.
Regarding (1) it was planned to study the faithfulness of the embedding of
further challenging quantified non-classical logics in HOL. The hypothesis has
been that in all cases the faithfulness, i.e. soundness and completeness of the
embedding, can be shown when a notion of Henkin semantics is assumed (on
the side of both logics). Regarding (2), a close collaboration with the projects
Leo-II and Leo-III, running in parallel at Freie Univerit¨at Berlin, was foreseen,
in addition to the use of the proof assistants Isabelle/HOL and Coq. With re-
spect to (3), a focus has been on applications in theoretical philosophy, resp. in
metaphysics, since there is a particular need for very expressive non-classical
logics in this area. For example, hyper-intensional second order modal logic is
utilised as the starting point in Zalta’s Principia Logico-Metaphysica [42], and
similarly expressive logics are studied in prominent recent textbooks by Stal-
naker [43] and Williamson [44]. Unfortunately, however, there had not been
any attempts prior to the CompMeta project to implement and automate such
challenging logic formalisms in computer systems. Mainstream knowledge rep-
resentation formalisms in computer science and artificial intelligence, including
e.g. semantic web technologies, typically fail to deliver (not only) in this ap-
plication context due to their lack of expressivity. The CompMeta project, in
contrast, intended to address this gap and to contribute to the pioneering of
the new area of computational metaphysics, which has its roots in the work of
Zalta and colleagues at Stanford University [45, 46, 47]. For the appropriate
modeling of foundational philosophical ontologies and for the formal analysis of
challenge arguments in philosophy (and beyond) a suitably expressive modeling
and reasoning framework was obviously required. With respect to (4), the goal
has been to design and offer an interdisciplinary lecture course on computational
metaphysics, in which the active use of the CompMeta framework was intended
to play a central role in combination with the training of a new generation of
students to independently master the approach.

4. Results of the CompMeta Pro ject
The main results and highlights of the CompMeta project are summarised
in this section and references are given to the most important publications
stemming from the project.
4.1. Application Study I: Ontological Argument for the Existence of God
Different modern variants of the ontological argument for the existence of
God, one of the still vividly debated masterpiece arguments in metaphysics
(see e.g. Sobel’s textbook [48] and the references therein), have been rigorously
analysed on the computer in the course of CompMeta. These contributions,
many of which were achieved in close collaboration with Bruno Woltzenlogel-
Paleo, received a media repercussion on a global scale.3
In the course of the conducted experiments [49, 50, 51, 52], the theorem
prover Leo-II detected a previously unknown inconsistency in Kurt G¨odel’s
prominent, HO modal logic variant [53] of the ontological argument, while Dana
Scott’s amendment [54] of it was verified for logical soundness in the interactive
proof assistants Isabelle/HOL [15] and Coq [55].4In Fig. 4 the axioms causing
the inconsistency in G¨odel’s manuscript are highlighted (see also the discussion
in Sect. 4.6.1). This inconsistency, which was missed by philosophers, is ex-
plained in detail in two conference papers [51, 50]. Further relevant insights
contributed or confirmed by ATPs e.g. include the separation of relevant from
irrelevant axioms, the determination of mandatory properties of modalities, and
undesired side-implications of the axioms such as the modal collapse.5
Further variants of G¨odel’s axioms were proposed by Anderson, H´ajek and
Bjørdal [57, 58, 59, 60, 61, 62]. These variants have meanwhile also been for-
mally analysed, and ATPs have even contributed to the clarification of an un-
settled philosophical dispute between Anderson and H´ajek [63]. In the course
of this work, different notions of quantification (actualist and possibilist) have
been utilised and combined within the semantical embedding approach [64].
Moreover, the modal collapse, whose avoidance has been the key motivation for
the contributions of Anderson, Bjørdal and H´ajek (and many others), has been
further investigated [65].
A significant further contribution has been achieved by David Fuenmayor,
a philosophy student recruited from in the computational metaphysics lecture
3See e.g. http://www.spiegel.de/international/germany/scientists-use-computer-to-
mathematically-prove-goedel-god-theorem-a-928668.html
4Scott was not aware of the inconsistency in G¨odel’s variant. Amongst others, he slightly
modified G¨odel’s definition of essence, which causes the inconsistency. Scott did so because
it felt natural to him to require that essential properties of an individual should actually be
possessed by that individual, and so he added a respective conjunct to the definition; cf. also
[50, 51].
5The modal collapse [56, 48] is a sort of constricted inconsistency at the level of possible
world semantics. The assumption that there may actually be more than one possible world
is refuted; this follows from G¨odel’s axioms as the ATPs quickly confirm. In other words,
G¨odel’s axioms, as a side-effect, imply that everything is determined (we may even say: that
there is no free will).

Figure 4: The axioms causing the inconsistency in G¨odel’s modal logic variant of the ontologi-
cal argument for the existence of God are highlighted in blue. The inconsistency was detected
by the HO ATP Leo-II. (Disclaimer: Unpublished works of Kurt G¨odel are Copyright Insti-
tute for Advanced Study and are used with permission. All rights reserved by Institute for
Advanced Study)

course at Freie Universit¨at Berlin (see Section 4.5). Fuenmayor, in a student
project [66, 67], formalised the most relevant parts of Fitting’s [68] textbook
Types, Tableaus, and G¨odel’s God. This book develops another interesting
emendation of the ontological argument, which — similar to other recent works
— aims at preserving the overall conclusion (necessary existence of God), while
at the same time getting rid of the modal collapse. Fitting’s means to achieve
this is by modifying the foundational logical system. Instead of an extensional
HO modal logic he employs a more expressive intensional HO modal logic, which
enables a different, and as Fitting explains, more adequate interpretation of
e.g. the notion of positive poperties in G¨odel’s argument.
The so far mentioned studies only address a small portion of the entire
relevant literature on the ontological argument. By extending these studies, it
can be expected that many further issues in human refereed contributions can
be revealed. A follow-up project could thus try to develop an encompassing
map that rigorously distinguishes sound from unsound work in this area.
Summary of key insights. Variants of extensional and intensional HO modal
logics can easily be implemented in the SSE approach; a very good degree of
proof automation can be achieved this way, matching or exceeding the argumen-
tation granularity we typically find in human authored publications on this sub-
ject; flexible logic modifications and combinations are supported; the approach
is practically highly useful and it combines automated theorem proving with
model and countermodel finding (the latter well supports the detection of typos
and minor issues during the formalisation process); it has been demonstrated
how the approach supports a novel, experimental style of work in metaphysics.
4.2. Application Study II: Zalta’s Principia Logico-Metaphysica
Formalising and automating masterpiece rational arguments in philosophy
with the SSE approach on the computer is not trivial. However, it still leads
to comparably small corpora of axioms, lemmata and theorems, and, hence,
it does not provide reliable feedback on the scalability of the approach for
larger and more ambitious formalisations. For that reason another challenge
has been tackled in the CompMeta project: the Principia Logico-Metaphysica
(PLM) of Edward Zalta [42], which aims at a foundational logical theory for
metaphysics, mathematics and the sciences (PLM thus intends to subsume the
Principia Mathematica [69]). Zalta has chosen a hyper-intensional, relational
second-order modal logic S5 [70, 71] as the foundational logic for PLM. It has
thus been a challenge question for CompMeta, whether this non-trivial foun-
dational logic can still be suitably encoded and automated in the semantical
embedding approach. Besides hyper-intensionality, a particular challenge has
been to overcome the conceptional gap between the relational core of PLM
and the functional core of HOL, and to suitably handle the different strengths
of comprehension principles supported in both logics that assert the existence
of relations and functions (the use of unrestricted comprehension principles in
PLM causes undesirable paradoxes and inconsistencies [72]). And, of course, a
main challenge has also been to deal with the comparably large size of PLM

in relation to the small axiom sets as studied in the context of the ontological
argument.
The author’s initial attempts, conducted during an extended research stay
at Stanford University in 2015/16, to semantically embed PLM’s base logic in
HOL by following a pure proof theoretic approach were unsuccessful. Later in
2016, in the course of the computational metaphysics lecture course in summer
2016 at Freie Universit¨at Berlin (see below), Zalta in an invited presentation
then outlined some ideas towards a set theoretical semantics for PLM, which
were suggested to him by Peter Aczel. This set theoretic perspective on PLM
subsequently enabled the development of a suitable shallow semantical embed-
ding of PLM in HOL. It was in fact Daniel Kirchner, a mathematics student
recruited the lecture course, who took on the challenge within an MSc thesis
project at Freie Universit¨at Berlin. Kirchner has meanwhile succeeded in for-
malising the PLM in Isabelle/HOL by suitably adapting the SSE approach so
that it soundly covers the base logic of PLM [73].
Kirchner’s work contributes various novel ideas and tools, including the pro-
vision of powerful automation means for PLM at different, cross-linked levels of
abstraction. For example, he developed a direct, tactic-based theorem prover
for PLM in Isabelle/HOL, which, one-to-one, implements the proof theory of
PLM as developed by Zalta with pen and paper. This object-level theorem
prover for PLM is connected with the HOL meta-level in Kirchner’s work via the
specifically tailored shallow semantical embedding he developed, and this link
establishes an Isabelle/HOL-internal criterion, modulo expansion of the seman-
tical embedding, for the soundness of his novel prover. Further, similar provers
are provided by him at well-defined, intermediate expansion levels. Kirchner’s
architecture thus provides multiple options for proof automation, ranging from
the full expansion of the semantical embedding (combined with calls to off-the-
shelf reasoning tools integrated with Isabelle/HOL via the Sledgehammer tool)
to the more intuitive, one-to-one automation of the proof theory of PLM within
Kirchner’s new tactic-based theorem prover.
An unexpected, but key result of Kirchner’s work has been the discovery
of a paradox in PLM [74, 73] (in the spirit of Russel’s paradox [75] for Frege’s
logic of the Begriffsschrift [76]): a deeply-rooted and known paradox is reintro-
duced in PLM, respectively, in the abstract object theory underlying the PLM,
when the logic of complex terms is simply adjoined to the frameworks specially-
formulated comprehension principle for relations. Kirchner’s result constitutes
a new and important paradox, given how much expressive and analytic power
is contributed by having the two kinds of complex terms in the system. The
results also provide a fresh perspective on the question of whether relational
type theory or functional type theory better serves as a foundation for logic and
metaphysics [72].
In close collaboration and supported by further experiments with Isa-
belle/HOL, possible emendations of PLM are currently being studied by Zalta
and Kirchner. The ongoing style of interaction well illustrates a new dynam-
ics in the scientific discovery process in metaphysics: rigorous experimentation
with implementations of foundational logical systems may quicken and inspire

the scientific discovery process in this area and also foster more reliable results.
Summary of key insights. The semantic embedding approach scales for ambi-
tious and large projects in metaphysics such as PLM; the approach is practically
applicable and already shows a good degree of automation, which will naturally
further improve (with the ATPs it relies upon); with the help of the implemented
framework new knowledge has been contributed; moreover, students can be well
motivated when using the approach to dive into complex, foundational questions
on the edge of current research in metaphysics in short time.
4.3. Application Study III: Free Logic and Axioms Systems for Category Theory
Partiality and undefinedness are prominent challenges in various areas of
mathematics and computer science. Unfortunately, however, modern proof as-
sistant systems and ATPs based on traditional classical or intuitionistic log-
ics provide rather inadequate support for these challenge concepts. Free logic
[77, 78, 79, 80] offers a theoretically appealing solution, but it has been consid-
ered as rather unsuited towards practical utilisation.
In collaboration with Dana Scott, a shallow embedding of free logic in HOL
has been developed and implemented in the CompMeta project. Just as for
the embeddings mentioned above, various state-of-the-art FO and HO ATPs
and model finders, which are integrated (modulo suitable logic translations)
with Isabelle/HOL via the Sledgehammer tool, can now be utilised to automate
reasoning in free logic. As a result we obtain an elegant and powerful imple-
mentation of an integrated interactive-automated theorem proving (and model
finding) environment for free logic.
To demonstrate the practical relevance of this new system, a series of axioms
systems for category theory has been systematically explored [81, 82, 83]. The
starting point has been a generalisation of the standard axioms for a monoid
to a partial composition operation. The purpose of this work has not been
to make or claim any contribution to category theory but rather to show how
formalisations involving the kind of logic required, in this case free logic, can be
implemented and validated within modern proof assistants such as Isabelle/HOL
when utilising the SSE approach.
Subsequently, the relation of the developed axiom systems to alternative pro-
posals from the literature has been studied within the framework. This includes
an axiom set proposed by Freyd and Scedrov in their textbook Categories, Al-
legories [84] for which we have revealed a technical flaw. Either all operations,
e.g. morphism composition, are total in their theory or their axiom system is in-
consistent. This observation applies when a free logic reading of their axiomatic
theory is adopted, where the free variables are assumed to range over all objects,
including the “undefined”. When adopting an algebraic reading of their axioms
system, where free variables range only over all defined objects, then strictness
axioms or conditions are missing in their framework. Both readings have been
formalised in the SSE approach.
Thus, in interaction with the SSE based implementation of free logic in
Isabelle/HOL, a minor (one may say technical) but nevertheless relevant issue

in a mathematics textbook has been revealed that domain experts had missed
before. The repair for this problem is quite straightforward, however. The
solution essentially corresponds to a set of axioms proposed by Scott [85] in the
1970s.
In the experiments reported above, the exploration studies have been signif-
icantly supported by series of experiments in which automated reasoning tools
were called from within the proof assistant Isabelle/HOL via the Sledgehammer
tool. Moreover, very useful feedback was obtained at various stages from the
model finder Nitpick [20], saving us from making several mistakes.
At the conceptual level this work exemplifies a new style of explorative
mathematics which rests on fruitful human-machine interaction with integrated
interactive-automated theorem proving technology. The conducted experiments
were such that the required reasoning was often too tedious and time-consuming
for humans to be carried out repeatedly with highest level of precision. It is
here where cycles of formalisation and experimentation efforts in Isabelle/HOL
provided significant support. Moreover, the technical inconsistency issue for
the axiom system of Freyd and Scedrov was discovered by ATPs, which further
emphasises the added value of automated theorem proving in this area.
Summary of key insights. The SSE approach is applicable also to free and in-
clusive logics, which so far were believed to be too difficult to automate and thus
of little practical relevance; quite to the contrary: as our experiments show, the
approach is indeed well suited for practical applications, e.g. for the exploration
of mathematical theories in domains such as category theory, where partiality
and undefinedness play a central role; new knowledge can be discovered this
way.
4.4. Theoretical Study IV: Universal Cut-Elimination
The development of cut-free calculi for expressive logics, such as quantified
non-classical logics, is usually a non-trivial task. However, for a wide range of
logics there exists a surprisingly elegant and uniform solution: simply utilise
the SSE approach. More precisely, by modeling and studying these logics as
semantically embedded fragments of HOL (with Henkin semantics), existing
cut-elimination results for HOL (with Henkin semantics) may be reused. In the
course of the CompMeta project, this idea has been further explored and exem-
plarily applied for proving cut-elimination for quantified conditional logics [86].
Conditional logics [87, 88], known also as logics of normality or typicality,
have many applications, including counterfactual reasoning, default reasoning,
deontic reasoning, metaphysical modeling, action planning and reasoning about
knowledge. Moreover, it is well known that they subsume normal modal logics,
since the modal box operator can be defined in terms of the more expressive
conditional operator. In contrast to the rather straightforward, Kripke-style
semantics of normal modal logics, conditional logics come e.g. with a HO selec-
tion function semantics, which makes them interesting objects of study. While
there is broad literature on propositional conditional logics, comparably few

authors have addressed FO extensions of conditional logics; those include Del-
grande [89, 90] and Friedman et al. [91].
The conditional logics studied in the course of the CompMeta project utilise
constant- and/or varying-domain FO quantifiers and they combine these with
further quantifiers for propositional variables. Such a rich combination has not
been adressed in the literature before. In particular, cut-elimination for these
logic(s) was still open (only for propositional conditional logics some related
results had been available [92, 93]; cf. also the references therein).
While earlier, practical work in the CompMeta project had already shown
that automation of quantified conditional logics is indeed feasible by utilising
the SSE approach [94], the second half of the project then switched the atten-
tion to the theoretical challenge of proving cut-elimination. It was then shown
[86], that by utilising the SSE approach for quantified (and non-quantified) con-
ditional logics, the question whether cut-elimination holds for them can in fact
be reduced to proving the faithfulness of their semantical embedding in HOL.
The latter task, however, constitutes a much simpler problem than proving
cut-elimination directly.
The exploited reduction principle is similarly applicable to other object logics
in the SSE approach, including many logics for which cut-elimination is still
open. However, special attention has to be payed to cut-simulation [86, 95],
which may render cut-elimination as a pointless criterion.
Key insights:. Cut-elimination of a given object logic can often be reduced to
showing the faithfulness of a shallow semantical embedding of this logic in HOL
(with Henkin semantics); the approach has been applied to prove cut-elimination
for some variants of quantified conditional logics, for which the question was still
open; it should be possible to obtain similar cut-elimination results for many
other challenging object logics by adopting the same reduction principle.
4.5. Educational Study V: Lecture Course on Computational Metaphysics
The early successes in the CompMeta project inspired the design of a world-
wide new lecture course on computational metaphysics [96, 97]. This lecture
course, which was set-up and held in collaboration with Alexander Steen and
Max Wisniewski, was awarded with the 2015/16 central teaching award of
Freie Universit¨at Berlin.6The course received substantial support from Jasmin
Blanchette (Amsterdam), Wolfgang Lenzen (Osnabr¨uck), Bruno Woltzenlogel-
Paleo (Canberra) and Edward Zalta (Stanford), who all contributed invited
guest lectures.
Students with heterogenous knowledge backgrounds from computer science,
mathematics, philosophy and physics attended the lecture course, and they came
from all three major universities of Berlin: Freie Universit¨at Berlin, Technical
6cf. http://www.fu-berlin.de/campusleben/lernen-und-lehren/2016/160428-lehrpreis/

University Berlin and Humboldt University Berlin. The attendance in the lec-
tures usually varied between 40 and 70 students. 36 students were formally
registered for the course and were graded.
The steep learning curves of nearly all students were astonishing, in partic-
ular in the second half of the course, when small, heterogeneous student groups
were formed to work each on an encoding and formal assessment of a different
publication in philosophy or mathematics by adapting and utilising the SSE
approach within the Isabelle/HOL proof assistant. The heterogeneous group
compositions, the 24/7 feedback from the Isabelle/HOL environment, and the
motivating project topics were prime reasons, as the author believes, for the very
good overall results of the course. A selection of project results has meanwhile
been presented at conferences or published as book chapters or journal articles
[98, 66, 74, 73, 99, 100, 67, 101]. Several students picked up follow-up topics
and turned them into BSc or MSc thesis projects [102, 103, 104, 105, 106].
A key ingredient for the successful implementation of the course has been,
that a single methodology and overall technique (the SSE approach) was used
throughout, enabling the students to quickly adopt a wide range of different logic
variants in short time within a single proof assistant framework (Isabelle/HOL).
The interdisciplinary course concept appears well suited to foster a much im-
proved logic education across disciplines.
Summary of key insights. The SSE approach is well suited to support a novel
form of university level logic education to heterogeneous groups of students;
excellent learning curves are possible; new teaching methods are enabled in
interaction with ITPs and ATPs for HOL.
4.6. Further Results and Comments
A range of related application studies (contributed partly also by collabora-
tors) has not been mentioned above. Amongst others, these works include se-
mantical embeddings of multivalued logic SIXTEEN [107], nominal logics [108],
temporal logics [109], paraconsistent logics, intuitionistic modal logics, etc. Also
the (partly ongoing) work of Streit, another student recruited from the compu-
tational metaphysics lecture course, on the formalisation of Boolos’ textbook
on provability logic [110] and on the formalisation of Bostrom’s simulation ar-
gument [111] has not been addressed above.
A relevant and challenging future application direction of the semantical
embedding approach lies in the modeling of legal, ethical, social and cultural
norms in intelligent machines [112]. To enable such applications, the author is
currently, in a collaboration with Leon van der Torre and Xavier Parent from
the University of Luxemburg, adapting the semantical embedding approach to
cover recent developments in the area of deontic logics [113]. Standard deon-
tic logic, which is just a normal modal logic, is obviously already covered by
the approach. More challenging has been the semantical embedding of e.g. in-
put/output logic [114] and dyadic deontic logic [115]. First results in this ap-
plication direction are promising [116, 117, 118, 119].

4.6.1. Note on Invention and Creativity in ATPs
As reported above, the theorem prover Leo-II detected the inconsistency of
the axioms in G¨odel’s original variant of the ontological argument; this incon-
sistency was not known to philosophers before. The clue in the proof of falsity
from the axioms [51, 50], is the empty essence lemma: from G¨odel’s [53] def-
inition of essential properties (essence, cf. Ess. in Fig. 4) it follows that the
empty property, i.e., the everywhere false property (alternatively we may pick
the property of being self-different), is an essential property of every individual.
Dana Scott [54] slightly modified G¨odel’s definition of essence in his variant
of the ontological argument (for cosmetic reasons — the inconsistency was not
known to him at the time), with the effect that the empty essence lemma is no
longer valid.
In its successful, automatic discovery of the inconsistency, the Leo-II prover
had to guess the instantiation of the empty property for a second-order variable
during proof search [51, 50]. This part in the proof is non-analytic: inspection
shows that one cannot synthesise the required instantiation by e.g. unification
with existing information (terms) in the search space. In fact, blind guessing of
this instantiation seems unavoidable here. The author considers this as a small
but nevertheless very interesting example for a true discovery (based on guessing
and checking) by an ATP; and this discovery is philosophically relevant.
4.6.2. Improved Infrastructure for HO Interactive and Automated Reasoning
In the course of the above works, and in close collaboration with de-
velopment of Leo-III, the CompMeta project fostered the development of a
reusable theorem proving infrastructure for a range of non-classical logics
[120, 121, 122, 123, 18]. This includes various, reusable encodings in Isa-
belle/HOL syntax, Coq syntax and in TPTP syntax, which have been all made
publicly available.7Moreover, this includes a flexible pre-processing module
[18, 124, 125] for the Leo-III prover (and any other TPTP THF [34] compli-
ant prover). This preprocessor turns Leo-III into a flexible reasoner for a very
wide range of propositional and quantified modal logics. In fact, no other im-
plemented system is available today which covers a wider range of modal logic
variants than Leo-III, and this approach can easily be extended for many other
non-classical logics that have been mentioned above.
5. Conclusion
The presented reasoning framework based on shallow semantical embeddings
in HOL constitutes the most widely applied universal logical reasoning approach
available to date. There is, however, a significant difference to Leibniz’ origi-
nal idea of a characteristica universalis and to various related proposals in the
literature. Instead of proposing a single, universal object-level formalism, the
7Cf. e.g. https://github.com/FormalTheology/GoedelGod

shallow semantical embedding approach supports many different competing ob-
ject logics from the logic zoo. No ontological commitment is enforced at the
object logic level. For example, the approach well supports both classical and
intuitionistic object logics, and can even elegantly combine them. The concrete
portfolio of embedded object logics is determined by the specific requirements of
an application at hand. Only at meta-level a single, unifying logic is provided,
namely HOL (or any richer logic incorporating HOL). By unfolding the defini-
tions of the logic embeddings, problem encodings utilising these object logics
are uniformly mapped to meta-logic HOL. This way Leibniz’ vision of a char-
acteristica universalis is realised in an indirect way in the presented approach:
universal logical reasoning is established (only) at the meta-level in HOL.
The presented universal logical reasoning framework has many challenging
applications in artificial intelligence, computer science, philosophy, mathematics
and natural language processing. A most relevant and timely application di-
rection concerns the application of the semantical embedding approach for the
modeling of ethical, legal, social and cultural norms in intelligent machines [116],
ideally in combination with the realisation of human-intuitive forms of rational
argumentation in machines complementing internal decision making means at
the level of statistical information and subsymbolic representations.
Acknowledgements. A big thanks (in alphabetical order) goes to all collabo-
rators and supporters of the CompMeta project, including (but not limited
to): Jasmin Blanchette, Harold Boley, Frode Bjørdal, Chad Brown, Maximilian
Claus, Ali Farjami, David Fuenmayor, Tobias Gleissner, Max Haslbeck, Daniel
Kirchner, Hanna Lachnitt, Wolfgang Lenzen, Tomer Libal, Irina Makarenko,
Paul Oppenheimer, Jens Otten, Xavier Parent, Larry Paulson, Florian Rabe,
Raul Rojas, Fabian Sch¨utz, Hans-J¨org Schurr, Dana Scott, Alexander Steen,
David Streit, Geoff Sutcliffe, Leon van der Torre, Max Wisniewski, Bruno
Woltzenlogel-Paleo, Edward Zalta, Marco Ziener.
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